## 1. (15 points)

The linear combination of vectors $v^T=[1,2,0]$ and $w^T=[0,0,1]$ fill a $plane$ in $R^3$.

1. Describe that plane with in matrix form.
1. Find one vector **NOT** that plane.
1. Find a vector perpendicular to that plane.

## 2. (10 points)

I. A vector $v = (1,1,\dots,1)$ in 9-dimensions, please find a unit vector $\hat{v}$ in the same direction as $v$.
II. Find another unit vector $\hat{w}$ that is perpendicular to $v$.

## 3. (40 points) A system of linear equation as:
$$
\begin{aligned}
x_1 + x_2 + x_3 &= 7 \\
x_1 + x_2 - x_3 &= 5 \\
x_1 - x_2 + x_3 &= 3
\end{aligned}
$$

1. Use elimination to make $PA \rightarrow U$ and back substitution to solve $x$.
1. Factorize $PA = LU$.
1. Compare the multiplier of each elimination step and lower triangle matrix $L$.
1. Using Gauss-Jordan to find $A^{-1}$ and make $x = A^{-1}b$.

## 4. (20 points)

If a $3\times3$ matrix $A$ conducts this operation: `row1 + row2 = row3`,

* Is $A$ singular or invertible?
* Find a nonzero solution of $Ax = [0,0,0]^T$.

## 5. (15 points)

* The row vector $x^T$ times $A$ times the column $y$ produces what number?

$$
x^T A y =
\begin{bmatrix}
0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6
\end{bmatrix}
\begin{bmatrix}
0\\
1\\
0
\end{bmatrix}
$$

- This is the row $x^T A = \_\_\_\_\_\_\_\_\_\_$ times the column $y$.
- This is the row $x^T = [0,1]$ times the column $Ay = \_\_\_\_\_\_\_\_\_\_$.